Solveeit Logo
Login

Question

Question: Find the integral \(\int {\dfrac{{\cos e{c^2}x}}{{(1 - {{\cot }^2}x)}}} dx\). A) \(\dfrac{1}{2}\lo...

Find the integral ∫cos⁡ec2x(1−cot⁡2x)dx\int {\dfrac{{\cos e{c^2}x}}{{(1 - {{\cot }^2}x)}}} dx.
A) 12log⁡∣1+cot⁡x1−cot⁡x∣+C\dfrac{1}{2}\log \left| {\dfrac{{1 + \cot x}}{{1 - \cot x}}} \right| + C
B) −12log⁡∣1+cot⁡x1−cot⁡x∣+C - \dfrac{1}{2}\log \left| {\dfrac{{1 + \cot x}}{{1 - \cot x}}} \right| + C
C) 12log⁡∣1−cot⁡x1+cot⁡x∣+C\dfrac{1}{2}\log \left| {\dfrac{{1 - \cot x}}{{1 + \cot x}}} \right| + C
D) None of these

Explanation

Solution

To solve the given integral we can use substitution method. Substitute for cot⁡x\cot x and express other functions in terms of that. Then applying the necessary integral formulas we get the result.

Formula used: For any variable x,x, ddxcot⁡x=−cos⁡ec2x\dfrac{d}{{dx}}\cot x = - \cos e{c^2}x
∫1a2−x2dx=12alog⁡∣1−x1+x∣+C\int {\dfrac{1}{{{a^2} - {x^2}}}} dx = \dfrac{1}{{2a}}\log \left| {\dfrac{{1 - x}}{{1 + x}}} \right| + C, for any constant a, where CC is the constant of integration.
Here ddx(f(x))\dfrac{d}{{dx}}(f(x)) represents the derivative of the function f(x)f(x) respect to xx and ∫(f(x))dx\int {\left( {f(x)} \right)} dx represents the integral of the function f(x)f(x) with respect to xx.

Complete step-by-step answer:
To find the integral of cos⁡ec2x1−cot⁡2x\dfrac{{\cos e{c^2}x}}{{1 - {{\cot }^2}x}},
Put t=cot⁡xt = \cot x
Differentiating on both sides of the equation we have,
⇒dt=−cos⁡ec2xdx\Rightarrow dt = - \cos e{c^2}xdx, since the derivative of cot⁡x\cot x is −cos⁡ec2x - \cos e{c^2}x.
⇒cos⁡ec2xdx=−dt\Rightarrow \cos e{c^2}xdx = - dt
Substituting we get,
∫cos⁡ec2x(1−cot⁡2x)dx=∫−dt1−t2\int {\dfrac{{\cos e{c^2}x}}{{(1 - {{\cot }^2}x)}}} dx = \int {\dfrac{{ - dt}}{{1 - {t^2}}}}
We know,
∫1a2−x2dx=12alog⁡∣1−x1+x∣+C\int {\dfrac{1}{{{a^2} - {x^2}}}} dx = \dfrac{1}{{2a}}\log \left| {\dfrac{{1 - x}}{{1 + x}}} \right| + C, for any constant a, where CC is the constant of integration.
Using this equation here as t=cot⁡xt =\cot x , a=1a = 1
⇒∫cos⁡ec2x(1−cot⁡2x)dx=∫−dt1−t2=−12×1log⁡∣1+t1−t∣+C\Rightarrow \int {\dfrac{{\cos e{c^2}x}}{{(1 - {{\cot }^2}x)}}} dx = \int {\dfrac{{ - dt}}{{1 - {t^2}}}} = - \dfrac{1}{{2 \times 1}}\log \left| {\dfrac{{1 + t}}{{1 - t}}} \right| + C
⇒∫cos⁡ec2x(1−cot⁡2x)dx=−12log⁡∣1+t1−t∣+C\Rightarrow \int {\dfrac{{\cos e{c^2}x}}{{(1 - {{\cot }^2}x)}}} dx = - \dfrac{1}{2}\log \left| {\dfrac{{1 + t}}{{1 - t}}} \right| + C
Substituting for t on the above equation we get,
⇒∫cos⁡ec2x1−cot⁡2xdx=−12log⁡∣1+cot⁡x1−cot⁡x∣+C\Rightarrow \int {\dfrac{{\cos e{c^2}x}}{{1 - {{\cot }^2}x}}} dx = - \dfrac{1}{2}\log \left| {\dfrac{{1 + \cot x}}{{1 - \cot x}}} \right| + C

So, the correct answer is “Option B”.

Additional Information: Integration and differentiation are two main operations of Calculus. One is the inverse of the other. Integration has many applications such as area between curves, volume, probability etc and in Physics as well.

Note: There are different methods to solve an integral. Here we choose the substitution method because of the relation between cot⁡x\cot x and cos⁡ec2x\cos e{c^2}x. In case of definite integrals, where the differentiation has to be done within the limits, when we change the variable, we have to be careful to apply the corresponding changes to the limits too. Constant integration is important when writing the integral value, because the constant term has always derivative zero.